Neue Entdeckungen. [Contruction of the regular 17-gon.] Col. 554 in: Intelligenzblatt der Allgemeinen Literatur-Zeitung, No. 66, 1 June, 1796. GAUSS'S FIRST PUBLISHED PAPER

GAUSS, Carl Friedrich

Published by Jena, 1796

First edition of the 19-year-old Gauss's first published paper, his announcement of his discovery that the regular polygon with 17 sides is constructible using ruler and compasses. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which regular n-gons (that is polygons with n edges) are constructible and which are not? Gauss's discovery that the regular 17-gon is constructible was the first step forward on this question since the Greeks. Only five years later, in his 'Disquisitiones arithmeticae' (DA), Gauss gave the complete solution of the problem (p. 637), but this first discovery "was one of which Gauss was vastly proud throughout his life and also, according to Sartorious von Waltershausen, the one which decided him to dedicate his life to mathematics . . . according to Weber, Gauss requested that the regular polygon of seventeen sides should be engraved on his tombstone. In a letter to Gerling dated January 6, 1819, Gauss elaborates the thought underlying the theory of polygons in his DA, through special reference to the polygon of seventeen sides; then after a paragraph generalizing the result to suitable [Fermat] prime numbers p = 3, 5, 17, 257, 65537, . . . he continues: 'The history of this discovery has up to the present nowhere been publicly alluded to by me; I can give it very exactly, however. The day was March 29, 1796 [when Gauss was 18] and chance had absolutely nothing to do with it. Before this, indeed during the winter of 1796 (my first semester in Gottingen), I had already discovered everything related to the separation of the roots of the equation (x^p-1)/(x-1) = 0 into two groups, on which the beautiful theorem on the lower part of page 637 depends, without making note of the day. After intensive consideration of the relation of all the roots to one another on arithmetical grounds, I succeeded during a holiday in Braunschweig, on the morning of the day alluded to (before I had got out of bed), in viewing this relation in the clearest way, so that I could immediately make special application to the 17-side and to the numerical verification. Of course still other investigations of the seventh section of the DA were added later. I announced this discovery in the Literaturzeitung of Jena where my advertisement was published in May or June 1796'" (Archbald, 'Gauss and the Regular Polygon of Seventeen Sides,' The American Mathematical Monthly 27 (1920), pp. 323-326). Gauss's paper is followed by a short note by his teacher Eberhardt August Wilhelm von Zimmermann. Contained in the complete volume of the Intelligenzblatt for 1796, 4to, pp. [ii], [768], 48 (index), main text printed in double columns numbered 1-1536. Contemporary boards, paper label on spine (a bit rubbed, library release stamp on upper board and front paste-down). A copy of this volume sold at a French auction in 2020 for 1065 euros, but we are not aware of any other copy having appeared on the market. Seller Inventory # ABE-1603894005485

Bibliographic Details

Title: Neue Entdeckungen. [Contruction of the ...
Publisher: Jena
Publication Date: 1796
Binding: Hardcover
Condition: Very Good
Edition: 1st Edition